{
  "abstract": "Learning Gaussian mixture models (GMMs) using the Expectation-Maximization (EM) algorithm and its gradient-based variants is a fundamental problem in machine learning. It is known that randomly initialized (gradient) EM fails to learn multi-component GMMs in the exact-parameterized setting, where the number of components matches that of the ground-truth GMM. Recently, global convergence of gradient EM has been established in the over-parameterized setting, where more components are used, provided that the ground-truth components are well separated. In particular, the minimum separation between ground-truth components is required to scale as $Ω(\\sqrt{d})$, where $d$ is the dimension. In this paper, we show that this dimensional dependence is unavoidable in high-dimensional settings. Specifically, we consider a hybrid EM algorithm that uses standard EM updates for the mixing weights and gradient EM updates for the component means. For any $ε> 0$, we prove that when the dimension is sufficiently large, in the worst case a separation of order $Ω(d^{0.5-ε})$ is insufficient to guarantee global convergence of population gradient EM in sub-exponential time under random initialization, even in the over-parameterized regime. Our result establishes an almost optimal worst-case lower bound on the ground-truth separation required for learning Gaussian mixtures via gradient EM in high dimensions.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2610.07551",
  "author_total": 5,
  "authors": [
    {
      "name": "Yiran Zhang"
    },
    {
      "name": "Mo Zhou"
    },
    {
      "name": "Weihang Xu"
    },
    {
      "name": "Maryam Fazel"
    },
    {
      "name": "Simon S. Du"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/8egqb6ejy63pj.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2610.07551",
  "fulltext_url": null,
  "id": "8egqb6ejy63pj",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2610.07551",
  "language": null,
  "licence": "LicenseRef-arXiv-nonexclusive-1.0",
  "licence_label": "arXiv licence",
  "licence_source": "article",
  "licence_url": "http://arxiv.org/licenses/nonexclusive-distrib/1.0/",
  "metadata_licence": "CC0-1.0",
  "native_subjects": [
    [
      "arxiv",
      "cs.LG"
    ],
    [
      "arxiv",
      "stat.ML"
    ],
    [
      "openalex",
      "1702|Bayesian Methods and Mixture Models"
    ]
  ],
  "openalex": "W7220909375",
  "pdf_url": "https://arxiv.org/pdf/2610.07551",
  "pmcid": null,
  "pmid": null,
  "published": "2026-10-06",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/8egqb6ejy63pj.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:27:32Z",
      "native_id": "2610.07551",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2610.07551"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:stat.ML",
      "subfield": "statistics"
    }
  ],
  "title": "Is $\\sqrt{d}$ Separation Necessary for Gradient EM to Learn Gaussian Mixtures in High Dimensions?",
  "title_text": "Is √d Separation Necessary for Gradient EM to Learn Gaussian Mixtures in High Dimensions?",
  "type": "preprint",
  "updated_at": "2026-10-30T12:27:32Z",
  "url": "https://research.draconitic.com/paper/8egqb6ejy63pj",
  "venue": "arXiv",
  "versions": []
}